Over the weekend I added some isolation to the charger and removed the ground connections, so the charger is completely floating now (relative to ground). This was done by putting some ABS plastic under the charger, and trimming off the ground wires that attach to the charger.
I also replaced the BMS master board, as the feedback from the charger might have damaged it. Not sure what did damage the BMS, but it is working now. I also had to turn the voltage down on the charger a bit – I suspect the voltage adjust knob is moving on its own.